Sure, let's factor the polynomial [tex]\(4x^5 - 20x^4 - 36x^3\)[/tex] step by step completely.
1. Identify the greatest common factor (GCF):
Look at the coefficients: [tex]\(4, -20, -36\)[/tex]. The greatest common factor of these numbers is 4.
Also note that each term contains at least [tex]\(x^3\)[/tex]. So, the GCF of all the terms is [tex]\(4x^3\)[/tex].
Factor out the GCF:
[tex]\[
4x^5 - 20x^4 - 36x^3 = 4x^3 (x^2 - 5x - 9)
\][/tex]
2. Factor the remaining polynomial [tex]\(x^2 - 5x - 9):
Check if it can be factored further. We look for two numbers that multiply to \(-9\)[/tex] (the constant term) and add up to [tex]\(-5\)[/tex] (the coefficient of the linear term).
In this case, there are no integer factors of [tex]\(-9\)[/tex] that sum to [tex]\(-5\)[/tex]. Hence, the quadratic polynomial [tex]\(x^2 - 5x - 9\)[/tex] cannot be factored over the integers.
Thus, the polynomial completely factored is:
[tex]\[
4x^3 (x^2 - 5x - 9)
\][/tex]
This is the fully factored form of the polynomial [tex]\(4x^5 - 20x^4 - 36x^3\)[/tex].
Final Answer:
[tex]\[
4x^3(x^2 - 5x - 9)
\][/tex]
